I would like to find death records for an American ancestor who died on a business trip to Paris in about 1840. What would be some good online sources or specific record sets to try?
Ancestry has "Reports of Deaths of American Citizens Abroad, 1835-1974" which is based on records held at NARA. NARA describes the records for the period you're interested as:
1835-1855. Record of Death Notices of U.S. Citizens Aboard (Inventory 15, Entry 848) in one volume, arranged chronologically, showing the date of a death notice sent by the Department of State to a U.S. newspaper for publication. Indexed with images online as part of the "Reports of Deaths of American Citizens Abroad, 1835-1974" database on Ancestry.com (available for a fee).
If you are only looking for the date, you can search online the City of Paris civil registry reconstituted books.
Select Acte de décès (Death record) for Type d'acte, enter the last name of your ancestor in the Nom de la personne recherchée field and press Rechercher (Search).
That should bring a page with a list of death records tables for people with last names equal or close to the one you entered, and for one or more periods.
Select the period(s) containing 1840 and that will open a page where you can browse all death records matching the name you gave. The information you'll get will be the year, the arrondissement (district), the last name, the first names, and the death date.
With this information, you should be able to get a more detailed death record but unfortunately, these records aren't available on line, only on site.